During the time that the Soviet Union existed, there was an official structure of government and an unofficial structure which held the true power. In essence, the Communist Party was the true government of the Soviet Union, regardless of what their constitution said.
Under the Soviet constitution, there was a legislative branch, an executive branch, and a judicial branch. The Supreme Soviet was the legislative branch and it was officially in charge of making laws. The Chairman of the Praesidium of the Soviet Union was the official head of state and head of the executive branch.
However, the real power in the Soviet Union rested with the Communist Party. The true ruler of the country was the General Secretary of the Communist Party. This was the post held, for example, by Stalin who had no official governmental post. It was the Party that dictated policy and had the true power over what was done in the Soviet Union.